IN35 · Ask atoms which way to tune

Optical Lattice Clock

What turns a narrow atomic line into a working clock? Atoms tell a laser which way to tune when successive probes on either side of a resonance excite different fractions. Feedback keeps the laser near the line centre. Independent measurements of environmental shifts are still needed to connect that centre to an unperturbed atomic frequency.

Interrogate trapped atoms on both sides of a resonance, retain classified atom counts and servo commands, and calibrate one thermal shift.

Which way should the laser move?

Drag the gold frequency dial. A changed pulse needs matching standards.

Servo command $0\,\mathrm{Hz}$

Atomic error estimate $0\,\mathrm{Hz}$

Calibrated static thermal shift $0\,\mathrm{Hz}$

Shift-corrected command $0\,\mathrm{Hz}$

Probe duration

T/\mathrm{s}

0.18

Feedback gain

g

0.55

Atoms per probe

N

400

Initial frequency command

u_0/\mathrm{Hz}

1.4

Environment, noise and simulation

Measured enclosure temperature

T_{\mathrm{env}}/\mathrm{K}

300

Dead time per probe

t_d/\mathrm{s}

0.12

LO frequency noise per probe

\sigma_{\mathrm{LO}}/\mathrm{Hz}

0.12

Classification error probability

\epsilon

0.015

Simulated unperturbed offset

\delta_{\mathrm{sim}}/\mathrm{Hz}

0.9

Acquisition seed

n

3501

The simulated offset generates the atoms only. Inference receives classified counts, frequency commands and independently acquired standards.

Model assumptions

Coherent two-level square pi pulses interrogate separately prepared finite ensembles. Readout records classified atom numbers, not fluorescence photon counts. Projection noise, symmetric classification error and independent shot-constant Gaussian local-oscillator frequency noise are declared. Preparation, readout and dead time are retained; no continuous laser-noise spectrum or Dick-effect calculation is supplied. A coarse prior must place the line in its central capture neighbourhood; insufficient central-line signal is rejected. Independent known detunings calibrate a local two-side discriminator; a withheld check is excluded. A separate synthetic frequency reference and measured homogeneous-enclosure temperatures calibrate a static fourth-power thermal shift over 280–320 kelvin. The coefficient is Sr-like teaching data; dynamic BBR, lattice, Zeeman, density, gravity and other shifts are omitted. The shift correction is conditional on these inputs, not a real clock uncertainty. Allan deviation uses overlapping averages of the latter half of retained servo-command offsets divided by an external nominal carrier. It measures command stability with the stated cycle sampling and no dead-time correction, not actual continuous optical output, absolute accuracy or a realized optical second. Atom size, lattice spacing and marker speeds are schematic; up to 36 markers represent the finite ensemble.

Retained two-side clock cycles

Cycle counts, calibration and timing

Last eight retained probe pairs; classified atoms, not photons
Pair	Low excited / total	High excited / total	Before / after (Hz)	Start (s)

Continue with Ramsey spectroscopy · the optical frequency ruler

Start with the essentials

Focus question: What turns a narrow atomic line into a working clock?
One-sentence intuition: Measure the slope using known frequency offsets, then convert each pair of counted excitation fractions into a command update.

Core mathematical model

Two readouts, one update

e_k=\frac{p_{+,k}-p_{-,k}-b}{S},\qquad u_{k+1}=u_k-g e_k

Subtract the calibrated intercept from the high-minus-low excitation fraction and divide by the measured local slope. Gain sets the fraction corrected per pair. Far from the line the discriminator loses direction.

Overlapping Allan variance of commands

\sigma_{y,\mathrm{cmd}}^2(mT_c)=\frac{1}{2(N-2m+1)}\sum_{j=0}^{N-2m}\left(\bar y_{j+m}^{(m)}-\bar y_j^{(m)}\right)^2

Use the uniformly spaced latter half of retained commands, normalized by the external nominal carrier. Differences between overlapping adjacent averages measure command fluctuations and cancel constant offsets. No confidence interval, dead-time correction or actual optical-output measurement is supplied.

A conditional thermal correction

\Delta\nu_{\mathrm{th}}=a\left(\frac{T}{300\,\mathrm{K}}\right)^4,\qquad u_{\mathrm{corrected}}=u-\Delta\nu_{\mathrm{th}}

Separate reference observations fit the fourth-power coefficient, excluding a withheld temperature. Apply the static correction only within the calibrated temperature interval and for a homogeneous enclosure.

Common difficulties

Does low Allan deviation establish accuracy?

Typical misconceptionA quiet command trace proves an absolutely accurate atomic frequency.

Better mental modelDifferencing adjacent averages cancels constant errors. This statistic describes servo commands; continuous output and omitted shifts require independent measurements.

Run the experiment

01
Follow the resonance
Watch classified atom counts and the next command. Drag the gold frequency dial on the apparatus.
What to observe: The excitation imbalance changes and feedback moves the command. Preview counts and atom colours update during the drag.
02
Reduce atom number
Choose the few-atoms demonstration and compare excitation fractions and commands.
What to observe: Binomial fluctuations increase when fewer atoms make each decision.
03
Change the pulse
Lengthen the pulse, retain a mismatched record, then acquire matching standards.
What to observe: The old calibration belongs to the old pulse. Inference rejects the mismatch and keeps the raw counts.
04
Separate stability from shift
Warm the enclosure and inspect independent thermal references and the corrected command.
What to observe: Temperature moves the whole resonance. Allan deviation is insensitive to a constant offset.

Optical Lattice Clock

Retained two-side clock cycles

From atom counts to a frequency command

Core mathematical model

Two readouts, one update

Overlapping Allan variance of commands

A conditional thermal correction

Common difficulties

Does low Allan deviation establish accuracy?

Run the experiment

Follow the resonance

Reduce atom number

Change the pulse

Separate stability from shift

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